And more solutions

Starting with Jer's y=x(+/-)x*sqrt(2016/x)

The (-) gives positive solutions if x = 2016*n^2/m^2 where m = 1,2,4,3,6 or 12 (because sqrt(2016) = 12*sqrt(14)

Let z = n/m

Then x = 2016(z^2)

y = 2016(z^2 - z)

z = 2016(z^2 - 2z) = 2016z(z-2)

In order for z to be positive, z must be > 2

The smallest z that therefore works is 25/12

z = 25/12 yields 2016*(625/144, 325/144, 25/144) = (8750, 4550, 350).

The Geometric progression is 8750,4550,2366, with a ratio of 13/25 (a.k.a., 1 - 1/z)

Total x + y + z = 13650, so this does not beat any entries on Jer's list, the last entry of which is 12096